Nonlinear Precomputed Radiance TransferPaul Green, Jan Kautz, Wojciech Matusik, Fredo Durand & Henrik Wann JensenFigure 1: A highly glossy object illuminated under two different environments. Notice the interreflections of the tail on the head (left image) and foot (right image). IntroductionWe propose a realtime method for rendering static objects with complex materials under distant allfrequency lighting. Existing precomputed light transport approaches [1,2] can render objects with complex material properties (e.g., anisotropy, translucency) and complex global illumination effects (e.g., interreflections, soft shadows, subsurface scattering). However, they are either restricted to lowfrequency illumination or they assume a fixed viewer. We propose a solution that allows both. Our method is based on two components: (1) a compact nonlinear representation for precomputed light transport that can be integrated rapidly with the distant allfrequency illumination, and (2) a new interpolation scheme that enables rendering of an arbitrary viewpoint from only a sparse set of precomputed views. Figure 1 shows an object rendered using our method. AlgorithmWe can view the scene as a "black box" linear system that transforms the input radiance into the output radiance. This linear system can be written as: where x is a point on the surface, wo is the outgoing direction, wi is the incoming direction, Li is the vector of (distant) incident radiance, T() is the light transport kernel (represented as a vector, with a coordinate for each direction wi), and Lo is the exitant radiance [Ng et al. 2003]. Both incident radiance Li and the transport kernel T() are sampled over the sphere of incoming directions. T() captures the complex transport effects occurring in the scene. We represent the transport vector with a nonlinear approximation. T() is represented as a sum of constantvalued box functions (1 inside the box and 0 outside the box) of arbitrary size and position: where Kj defines the size and position of box j and wj is a scalar weight of the box. This representation has two advantages: an efficient integration algorithm and a simple interpolation technique that allows us to approximate T(x,w'o ) for an arbitrary (i.e. not precomputed) direction w'o. Using this representation, we can approximate Equation 1 efficiently: Using a summed area table representation for Li, we can compute < Kj, Li> quickly. For an arbitrary outgoing direction, w0o , it is not sufficient to interpolate the resulting Lo from the nearest precomputed view directions because of possible highfrequency viewdependent components (e.g., specular reflections). Instead, we synthesize a new light transport vector T(x,w'o ) by interpolating the box parameters (Kj and wj) from the three nearest precomputed directions. The synthesized transport vector is a barycentric combination of the three nearest transport vectors. T(x,w'o ) is then integrated with the incident lighting (Equation 3) to produce the exitant radiance Lo(x,w'o ) at position x along direction w'o . Research SupportThis research was supported by a Ford Foundation Graduate Fellowship, MIT Presidential Fellowship, NSF CAREER award 0447561 "Transient Signal Processing for Realistic Imagery," NSF CISE Research Infrastructure Award (EIA9802220), INRIA equipe associee and the MITFrance Program. References[1] R. Ng R. Ramamoorthi and P. Hanrahan. Allfrequency shadows using nonlinear wavelet lighting approximation. In ACM Transactions on Graphics 22,3 (July) pp. 376381. [2] P Sloan, J. Kautz and J. Snyder. Precomputed radiance transfer for realtime rendering in dynamic, lowfrewuency lighting environments. In ACM Transactions on Graphics 21,3 (July) pp. 527536. 

